1

ياعزيزهللا

Dynamic shakedown and degradation of elastic

reactions in laterally impacted steel tubes

M. Zeinoddini*,1

and G.A.R. Parke**,2

* Faculty of Civil Engineering, KNToosi University of Technology, Tehran, Iran..

**Department of Civil Engineering, University of Surrey, UK.

____________________________________________________________________________________

ABSTRACT

This paper reports some interesting observations on the response of laterally impacted steel

tubes which in some respects have been considered to undergo an elastic shakedown. In an

experimental study on the behaviour of axially compressed tubes under lateral impacts, it has

been noticed that, after full development of plastic deformations in the impacted bodies, the

structural system ceases to exhibit additional plastic responses. The impacted tubes then exhibit

an elastic response. It has also been observed that the amplitude of the elastic excitations in the

specimens becomes more restricted as the load configuration moves close to a dynamic failure

state. With load conditions quite close to the dynamic failure limit, almost no elastic excitation

has been perceived from the impacted specimens. Additional numerical and analytical

investigations have been carried out on impacted tubes, frames and non-linear Single Degree of

Freedom (SDOF) systems and similar results have been obtained. Despite the non-cyclic nature

of the external loads in these impact cases, a phenomenon similar to elastic shakedown has

been observed.

Keywords: dynamic shakedown, elastic shakedown, steel tube, lateral impact, simulation,

dynamic failure, adaptation

NOMENCLATURE

1. Corresponding author. Tel:+98218770006, fax:+98218779476

Email address: zeinoddini@kntu.ac.ir

Faculty of Civil Eng., KNToosi University of Technology, ValiAsr-Mirdamad Cross, Tehran Iran.

2. G.Parke@surrey.ac.uk

2

A Acceleration

D Tube outer diameter

F Concentrated lateral load

Fo Dynamic lateral step load

Fp The minimum static concentrated lateral load required at mid-span of an

encastre tubular beam to produce a three hinge plastic collapse mechanism

(8D2t y/L)

K Stiffness of a spring in its linear range

L Tube length

M Lumped mass of a SDOF system

n Ratio between the applied step load and the plastic load of the spring

P Lateral push over load (Fig. 7)

Po Impact step load on the non-linear spring (Fig. 10)

Py Plastic load in the non-linear spring (Fig. 10)

Py Axial squash load of the tube (Dty)

t Tube wall thickness

t Time

u Displacement

stu Displacement under static load

.

u Velocity

..

u Acceleration

v Velocity

Radial frequency

y Material yield stress

1. INTRODUCTION

Classical ‘shakedown theory’ is related to the repetition of a quasi-static load in the structure and

is restricted to linear geometrical problems and elastic, perfectly plastic, materials. The

shakedown theory was promoted first by Bleich [1] and Melan [2] who gave the relevant criteria

3

for the ‘static theorem of shakedown’. Koiter [3] provided a solid and rational basis for the

theorem and proposed the so called ‘kinematics theorem for shakedown’. Other contributors

extended shakedown theory to more general structural and material models such as discrete or

discretised models, including hardening effects and geometric non-linearity effects [4].

This paper reports some observations on the response of laterally impacted steel tubes [5]

which in some respects have been considered to behave in a similar manner to elastic

shakedown. To categorise the findings, first classical, dynamic and pseudo-shakedowns have

been briefly reviewed. This is followed by experimental observations on the behaviour of

impacted tubes, results from numerical models of impacted tubes and frames along with the

response of a non-linear Single Degree Of Freedom (SDOF) system subjected to step loads

have been presented.

2. BACKGROUND TO SHAKEDOWN AND PSEUDO-SHAKEDOWN

2.1. Static Classical Shakedown

An elasto-plastic structure subjected to repeated cycles of (quasi-static) loads, varying within a

specific range, may eventually end in one of four typical states. In the first state, a purely elastic-

reversible response occurs and the deformations of the structures remain bounded within elastic

limits (Fig. 1.a). With the second state, some irreversible plastic deformation occurs in the

structure but the accumulated plastic dissipated energy in the whole structure (after each cycle

of loading) remains bounded. The structure eventually ceases to suffer further plastic

deformation and thus responds to subsequent cycles of loads in a purely elastic manner (Fig.

1.b). This behaviour is called ‘elastic shakedown’ or ‘adaptation’ [6].

If the amplitude of the load exceeds a threshold, either ‘plastic shakedown’ (third state of

behaviour) or ‘incremental collapse’ (fourth state of behaviour) occurs. With plastic shakedown

(also called alternating plasticity), the plastic energy after each cycle of loading still remains

bounded but the plastic strain increments change their sign during the loading process (Fig. 1.c).

Although with this kind of inadaptation, the plastic increment stays at zero in each cycle, local

material failure will occur due to low cycle fatigue.

With incremental collapse (called also ratcheting), the plastic strains increase cycle after cycle

so that, after a certain number of cycles, the net accumulation of plastic strains somewhere in

4

the structure will exceed the material ductility limit, or become unreasonably large for

serviceability. This behaviour is shown in Fig. 1.d and the structure is seen to be accumulating a

certain amount of energy during each cycle of loading which eventually leads to inadaptation of

the structure [7].

UNLOADING

LODING

DISPLACEMENT

LO

AD

a) Elastic cyclic response

UNLOADING

LOADING

DISPLACEMENTL

OA

D

b) Elastic shakedown or adaptation

UNLOADING

LOADING

DISPLACEMENT

LO

AD

c) Plastic shakedown or alternating

plasticity

UNLOADING

LOADING

DISPLACEMENT

LO

AD

d) Ratcheting or incremental collapse

Fig. 1: Four typical states of response in an elasto-plastic structure subjected to repeated cyclic

loads, varying within a specific range.

There have been a number of practical observations of different types of shakedown in structural

components under complex variable or cycling loadings. Haldar et al. [8] reported that interaction

of gravity and cyclic loads in soil-foundation components of offshore structures resulted in

ratcheting settlements. Rosson and Boothby [9] noted the occurrence of elastic shakedown in a

masonry arch bridge subjected to over loading which had developed irreversible deformations.

Some nuclear reactor components [10, 11], silos [12], and structural components in turbines and

aircraft [7] and components also in metallurgical industries [13] could become subject to

shakedown. Field observations indicated that many pavements do in fact shakedown rather than

deform continuously [14].

5

2.2. Dynamic Shakedown

Researchers have extended the static characteristics of the Bleich-Melan theorem to dynamic

problems [15]. With this approach the concept of bounded total plastic energy, typical of quasi-

static shakedown theory, was used to discriminate shakedown from non-shakedown cases. The

basis of this approach was that, the total plastic energy dissipated within the structure might

actually diverge as a result of the continued application of quasi-static loads, or indefinite

repetition of cyclic dynamic loads. In another proposed approach for dynamic shakedown,

repeated excitations of the structure were addressed instead of repeated loads. With this

approach (called minimum adaptation time), the shortest time required for a continuous solid

body with an elastic-plastic material to shakedown to an elastic state was defined theoretically

[4].

Two classes of dynamic shakedown problems can be envisaged for the related load schemes

and shakedown criteria; namely, restricted dynamic shakedown, in which the load is a specified

load history of either finite or even infinite duration, and for which the adaptation time criterion is

the most appropriate; and unrestricted dynamic shakedown, in which the load is an unknown

sequence of short-duration excitations, and for which the classic bounded-plastic-work criterion

is the most appropriate [4].

2.3. Pseudo-Shakedown

Some researchers reported that a phenomenon known as pseudo-shakedown could occur in a

rigid, perfectly plastic rectangular plate which strengthens with the development of finite

displacements when subjected to repeated dynamic impact loads having a triangular pressure

time history [16, 17].

Pseudo-shakedown, which is different from classical shakedown, takes place in some structures

when the permanent deformations created in the system by the first dynamic impact are less

than the permanent deformations resulting from equivalent static loading.

3. ADAPTATION/DEGRADATION OF ELASTIC REACTIONS

In a number of experiments on steel tubes subjected to combinations of axial pre-compression

and lateral impacts, there have been three distinctive and interesting observations. With one of

the tubes, after development of plastic deformations, the tube ceased to respond with further

6

plastic deformation and predominantly exhibited elastic oscillations. With the second specimen,

the amplitude of the elastic oscillations became more restricted when the applied loads moved

closer to the dynamic failure limit. Very close to the dynamic limit load, the impacted tube only

presented plastic deformations with no perceptible elastic reactions and the amplitude of the

elastic oscillations almost decreased to zero. These observations have been supported by

additional numerical and analytical investigations which are also reported in this section.

3.1. Experimental Observations

In the test program, which was carried out by the authors, tubular steel specimens were initially

axially pre-compressed and then subjected to lateral impacts at their mid span. An overall view

of the test rig is shown in Fig. 2. The specimens had one fixed and one free sliding support. A

self-reacting system of disc springs was placed behind the free sliding support. These

compressive springs allowed a pre-defined, axial compression to be applied to the specimen.

They were also employed to maintain the compression loads during the axial shortening of the

tube which is bound to happen during lateral impacts [18]. The specimens, each one meter long,

were cut from 6-7m cold-drawn seamless tubes with a nominal outside diameter of 100mm and

wall thickness of 2mm. Plates of 150×150×16mm were welded to the ends of each specimen.

The specimens were instrumented, set up in the impact rig, axially pre-compressed and then

impacted at mid-span by a dropping striker. The mechanical properties of the tube material are

given in Table 1.

Table 1: Mechanical properties of the tubes material from tensile and stub-column tests.

E (kN/mm2) y (N/mm

2) u (N/mm

2) u /y (%) ε u ()

Tensile test 200 516 538 104.3 11200

Stub column test 189 481 526 109.3 -

The striker, with an adjustable weight of 15 to 50kg, was able to travel within the vertical guides

and hit the specimen at right angles to the tube axis (Fig. 2). The striker had a 90o toughened

knife-edge indentor. The head of the indentor was sufficiently rounded to avoid the occurrence

of local tearing in the specimen. In these impact tests, the velocity and mass of the striker were

kept constant (7m/s and 25.45kg respectively) but the axial pre-compressions were different.

The axial pre-compressions were varying within a range of 0, 25, 27, 50, 60, 65, 70, and 75% of

the specimen squash load (Py=Dty).

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During the experiment the tubes performed in an identical manner. When the axial pre-

compression exceeded 0.65Py, an instant dynamic failure was triggered in the specimen by the

first impact. In these cases, the striker caused a dent in the specimen which grew deeper. The

tube then buckled and moved downward in a ‘dog leg’ shape.

3

4

2

1

9

5

6 10

8

7

1- Striker

2- Central Tower

3- Sliding Support

4- Disc Springs

5- Load Cell

6- Hydraulic Jack

7- Tie Rods

8- Base Frame

9- Tubular Specimen

10- Fixed Support

Fig. 2: Schematic view of the impact rig for testing of axially pre-compressed tubes subject to

lateral impact.

When the axial pre-compression was less than 0.65Py, the specimen did not fail during the

impact tests and remained stable. In these cases the first impact caused permanent local dents

and dimples in addition to an overall bowing in the tube but the deformations remained limited.

The first impact was followed by a number of rebounds. This was because no attempt was made

to prevent the bouncing of the striker on the specimen after its first hit.

When the axial pre-compression was 0.65Py, the striker made a relatively deep dent in the

specimen but it virtually stopped at a certain point and no bouncing was observed.

Table 2 provides some general results from the tests and Fig. 3 shows an undamaged specimen

together with post impact views of some other specimens. More details on the experimental

programme can be found in Zeinoddini, et al. [5 and 19].

8

In addition to the useful data from the experiments on the behaviour of the impacted tubes

(Zeinoddini, et al. [5 and 19]), it was noticed that the specimens elastic reactions was reducing

as the loading configuration approached the dynamic failure conditions.

3.1.1. Rebounds characteristics

It was already mentioned that in the tubes which did not fail, the first impact was followed by a

number of rebounds. The rebounds were the outcome of elastic reactions from the impacted

specimen, which produced an upward initial velocity in the striker. The striker rose to a certain

height due to this initial velocity and fell again for the next hit. The numbers of perceptible

rebounds and the bouncing duration (the time interval between the striker separation upward

from the specimen until its next contact) are given in Table 2. The table also shows the striker

velocity recorded on its first touch with the specimen and the striker velocities in the succeeding

hits.

Table 2: Specimens definition and some general impact test results.

General data Rebounds Impact velocities Bounce duration

Name P/Py (%)

Test Result

Recorded Number

1st

(m/s) 2

nd

(m/s) 3

rd

(m/s) 6

th

(m/s) 1

st

(ms) 2

nd

(ms) 3

rd

(ms) 5

th

(ms)

PD0 75 Failed 0 7.014 0 0 0 0 0 0 0

PD1 0 Stable 15 7.006 2.620 1.810 0.730 514 355 249 143

PD2 27 Stable 9 6.998 2.410 1.650 0.690 473 324 226 135

PD3 50 Stable 7 6.995 1.840 1.030 0.405 361 202 157 79

PD4 60 Stable 5 7.012 1.215 0.850 0.135 238 167 106 26

PD5 62 Stable 3 7.002 0.560 0.375 0 110 74 51 0

PD6 70 Failed 0 7.006 0 0 0 0 0 0 0

PD7 65 Failed 0 6.988 0 0 0 0 0 0 0

PD8 25 Stable 8 6.991 2.495 1.715 0.705 490 336 232 138

PD9 0 Stable 14 7.004 2.590 1.790 0.720 508 351 253 141

Table 2 shows that, in each test, as the number of striker hits increases, the impact velocity and

the bouncing duration (corresponding to the striker rising height) decreased. This was apparently

caused by dissipation of the external input energy through the system damping.

From Table 2 it can also be noticed that the number of discernible rebounds decreased as the

axial compression increased. Similarly, with a certain impact number, for instance the second

impact, when the axial pre-compression increased, shorter bouncing duration (accordingly a

shorter striker rise) and consequently less impact velocity for the next hit were recorded.

9

Fig. 3: View of an undamaged specimen together with post impact views of PD1, PD2, PD3,

PD4, PD6 and PD7 specimens (in left to right order).

The rebounds were created by the elastic (flexural) reactions from the impacted specimens.

Therefore, reductions in the number of recorded rebounds and similar parameters such as

bouncing duration and rebound velocities (Table 2) indicate a degradation of the elastic reaction

from the tubes. With the tests listed in Table 2, all parameters such as the tube dimensions,

velocity and mass of the striker (say the external input energy) remained primarily constant and

just the axial compression varied. The striker itself was made as rigidly as practically possible,

so it can be concluded that the elastic reaction from the specimens was gradually decreasing

out as the axial pre-compression increased towards the dynamic failure limits.

3.1.2. Reduction in the amplitude of elastic oscillations

In Fig. 4 the time histories of the impact load for specimens with different levels of axial pre-

compression are given. This figure displays snapshots of the first impact duration. It should be

noted that the specimen with 70% axial pre-compression (PD6), failed as a result of the impact

test, but the other specimens remained stable.

10

0

10

20

30

0 5 10 15 20

P/Py=0%

P/Py=25%

P/Py=27%

P/Py=50%

P/Py=60%

P/Py=70%

TIME (ms)

IMP

AC

T L

OA

D (

kN

)

Fig. 4: Variation of the impact load during the first impact by the change in the specimens' axial

pre-compression.

With each specimen, Fig. 4 demonstrates two types of fluctuations. One nearly has a half sine

shape with its duration close to the specimen half period in its first natural mode of vibration

(exhibiting a bowing mode shape measured to be around 31.6ms). The second type of

fluctuation has a period around 2ms which is related to other modes of vibration, possibly in the

tube wall, in the striker or in the disk spring system.

The impact loads in Fig. 4 were recorded using load cells placed inside the striker. So, they

inevitably report the (elastic) reactions from the specimens imparted to the striker. The half sine

fluctuation, most probably, represents the flexural reactions from the specimens. As it can be

noticed the maximum impact load (or the tube elastic reaction) decreases as the axial pre-

compression increases (or in other words the load conditions move closer to the dynamic failure

status). Fig. 4. shows that the amplitude of the second type of oscillations also decreases as the

axial pre-compression increases particularly after the initial impact. The response of the failed

specimen appears almost as if it is free from these oscillations. These observations, once more,

underline that the elastic reactions from the impacted specimens reduced as the loading

configuration approached the dynamic failure limits.

11

3.1.3. Failure case

When the striker hit the specimen PD7, which had an axial pre-compression of 65%Py, the tube

was severely damaged by the first impact but nevertheless remained stable. Interestingly with

this specimen no distinct rebound occurred after the first impact. The striker remained virtually

stationary on the damaged specimen. After about 1.6 seconds from the first impact (a relatively

long time in the experiment time scale), the tube suddenly failed making a large noise.

The ways in which the tube with 65%Py axial pre-compression failed, firstly indicated that the

load combination was particularly close to the specimen’s exact minimum failure load. With this

particular load configuration, the tube remained briefly in a stable but critical condition. In this

critical condition, the structural system required only a slight additional perturbation to make the

specimen unstable. This additional effect could have been any change in the system, such as a

small stress relief in the axial springs.

Secondly, as mentioned earlier, after the first impact on PD7, the striker showed no distinct

rebound. Considering that the rebounds are the outcome of elastic reactions from the impacted

body, absence of distinct rebounds indicated that almost no elastic reaction existed in the tube to

force the striker up from the specimen. Lack of the elastic response in this case, yet again,

indicated that the elastic reactions of the impacted tubes decreased towards zero as the level of

the applied load approached the dynamic failure states.

It is worth noting that the response of specimen PD7 was a rare physical observation. Out of 10

experiments, quite by chance, the impact conditions for PD7 (the striker velocity, tube

conditions, pre-compression level, etc) coincided with the minimum dynamic failure load. It is

clear that in these kinds of physical experiments the possibility of coming across this precise

point is quite low. This is because the minimum failure load can be regarded as a point in a

space made by different variables such as the striker weight, shape, velocity, axial compression

etc. With merely a slight deviation from this point, the specimen response would have moved to

become either stable or in a collapse condition.

3.2. Numerical examinations

Degradation of elastic reactions and behaviour similar to the adaptation or elastic shakedown

have also been observed in numerical models of lateral impacts on tubular members. The

12

simulations have been carried out using the ABAQUS non-linear finite element program (Hibbit,

et al. [20]). An implicit direct integration dynamic approach based on Newark’s constant average

acceleration method has been employed to solve the non-linear equations of the motion in the

impact analysis.

In the numerical models reported in this section, the impact has been introduced as a lateral

dynamic step load applied at the mid-span of the tubes. The axial pre-compression has been

kept constant at 50%Py while the impact load has been varied. No structural damping has been

incorporated into the finite element models. The von Mises yield criterion has been utilised to

model the inelastic material properties. An elastic, perfectly plastic material property has been

used. No strain rate effects have been considered for the steel material.

3.2.1. Individual tubes

Individual tubes have been numerically studied under lateral impact loads. The encastre end

tubes have been modelled using twenty four shell elements (S4R) in the circumference and fifty

in the longitudinal direction. These models allow for local deformations. The tubes have also

been modelled using 20 beam elements (type PIPE31) which exclude local deformations. The

modelled tubular member has an outer diameter of 356mm, a wall thickness of 12.7mm and a

length of 5700mm with a yield stress of 350 N/mm2.

Figs. 5, and 6 show the numerical responses of the individual tubes under different dynamic step

lateral loads applied at their mid-span, with local deformations excluded or included. The plastic

load Fp (=8D2t y/L) corresponds to a static concentrated lateral load required at the mid-span of

an encastre tubular beam member to produce a three hinge plastic collapse mechanism. Time

histories of non-dimensional lateral displacement at the impact position in the mid-span of the

tubes are displayed in these figures.

Figs. 5 and 6 indicate that some responses have remained bounded. This indicates that the

corresponding structural systems remained dynamically stable. By increase in the lateral impact

load certain responses have become unbounded, indicating the occurrence of a dynamic failure

in the structural system as a result of the impact load.

Figs. 5 to 6 also show that with an increase in the level of lateral impact load, the mean

deformation in the stable responses has increased but after development of plastic

13

deformations, the asymptotic oscillations remained elastic. The amplitude of these elastic

oscillations (which have a frequency close to the main bowing natural frequency of the tube) has

become more restricted with an increase in the level of impact load. With responses close to the

dynamic limit load, the asymptotic oscillations have almost faded out.

0

0.1

0.2

0.3

0.4

0 50 100 150 200

0.400 Fp

0.3932 Fp

0.3928 Fp

0.3925 Fp

0.3920 Fp

0.390 Fp

0.365 Fp

0.340 Fp

0.215 Fp

TIME (ms)

DIS

PL

AC

EM

EN

T/R

Fig. 5: Numerical time histories of lateral displacement at the impact position in the individual

tubes subjected to lateral step loads at their mid-spans (local effects excluded).

0

0.2

0.4

0.6

0.8

0 50 100 150 200

0.400 Fp

0.284 Fp

0.283 Fp

0.282 Fp

0.277 Fp

0.270 Fp

0.245 Fp

0.215 Fp

TIME (ms)

DIS

PL

AC

EM

EN

T/R

Fig. 6: Numerical time histories of lateral displacement at the impact position in the individual

tubes subjected to lateral step loads at their mid-spans (local effects included).

14

3.2.2. Impacted frames

Benchmark and large scale tubular frames were tested under static push over loads (Fig. 7) by

other researchers (Nichols et al. [21] and Bolt et al. [22]). These tubular frames have been

numerically modelled by the authors to study the behaviour of laterally impacted frames. The

numerical models of the frame have shown very good correlation with the experimental results

under quasi static push over loading. 1

56

20

mm

35

6 / 1

2.7

/ 3

50

35

6 / 1

9.1

/ 3

50

169 / 4.5 / 350

P

5944mm

35

6 / 1

2.7

/ 3

50

169 / 4.5 / 290

169 / 4.5 / 290

169 / 4.5 / 290

169 / 9.5 / 390

Hinge Unit

169 /

7.65

/ 320

169 / 7.65 / 320

35

6 / 1

9.1

/ 3

50

169

/ 6.3

/ 32

0169 / 6.3 / 320

169 / 4.5 / 290

169

/ 4.5

/ 29

0169 / 4.5 / 290

169

/ 4.4

5 / 2

90

169

/ 4.6

/ 29

0

169

/ 4.6

/ 29

0

169 /

7.65

/ 320

169 / 7.65 / 320

169 / 6.3 / 320

610x229x140kg UB GR 43B Yield = 320N/mm2

358x368x202kg UB GR 43B Yield = 320N/mm2

Fig. 7: Elevation and properties of the tubular frame, used in the benchmarking exercise

(Nichols et al. [22]).

The figures given on

each member are D,

t and y of the tube

in mm, mm and

N/mm2 respectively.

15

0

0.2

0.4

0.6

0.8

0 500 1000 1500 2000

0.450 Fp

0.378 Fp

0.376 Fp

0.374 Fp

0.365 Fp

0.340 Fp

0.270 Fp

TIME (ms)

DIS

PL

AC

EM

EN

T/R

Fig. 8: Numerical time histories of lateral displacement in the impacted tubular frame (local

effects excluded).

Two identical numerical models of the benchmark frames have been developed. In the first

numerical model, all frame members have been modelled using up to 20 beam elements (type

PIPE31) for each member. Using beam elements excludes modelling local deformations. In the

second numerical model the impacted chord member has been modelled using twenty four shell

elements (S4R) in the circumference and fifty in the longitudinal direction but the remaining

members have been modelled with beam elements. This model allows for local deformations to

be considered in the impacted member. Both models have been loaded vertically which

produced an axial compression in the chord members equal to 50% of their squash load.

Degradation of elastic reactions and behaviour similar to dynamic shakedown have also been

observed in the benchmark frames when subjected to a lateral impact at mid-span of the upper

chord member (see Fig. 7). Once more, in both numerical models, with an increase in the level

of lateral impact load, the mean deformations increased but with responses close to the dynamic

limit points, the amplitude of the ensuing oscillations has become more restricted (Fig. 8). Fig. 9

shows a post impact view of the numerical model of the tubular frame (with local effects

included) when subjected to a dynamic failure.

16

Fig. 9: Post impact view of the numerical model of the tubular frame, subjected to a dynamic

failure as a result of lateral impact (local effects included).

3.3. Analytical Model

The response of a non-linear SDOF system to a dynamic step impact load has also been

analytically studied. The system is shown in Fig. 10a and consists of a lumped mass M and a

non-linear spring. A step load of Po is applied to the system (Fig. 10b). A simplified elastic-

perfectly plastic behaviour has been considered for the spring (Fig. 10c). With

yuutu 1)(0 the spring, having a linear stiffness of K, remains elastic. Beyond 1u the

spring is perfectly plastic but possesses an elastic retardation path. No damping is included in

the system and at t=0. it has a zero initial condition.

In phase one of the response, yuutu 1)(0 (or 0 1 t t ), the equilibrium equation is:

0..

oPKuuM 1

17

With zero initial conditions .0)0()0(.

uu , MK / and n=Po /Py, the solution for the

differential equation ( Eq. 1) is:

)]cos(1[)( 1 tnutu 2

At t t 1 :

11 /)( uKPtu y 3

From Eqs. 2 and 3:

)1

(cos1 1

1n

nt

4

12)( 11.

nutu 5

a) Dynamic model of the SDOF system. b) The applied dynamic step load.

c) Plastic incremental law for the non-linear spring.

Fig. 10: Characteristics of the non-linear SDOF system.

P o

K M

u

K

-Py

u2

P

K

u1=uy

Py

O

t

P

P0

O

18

Eqs. 4 and 5 indicate that the system only reaches the non-linear part of the behaviour if n0.5.

In that case in phase two, u u t u1 2 ( ) (Fig. 10b) or t t t1 2 the equilibrium equation is:

0..

oy PPuM 6

With initial conditions from Eqs. 3 and 5, the solution of the above differential equation is:

1)(12)(

2

1)( 1

2

1

2

1 ttnttn

utu 7

The motion in phase two proceeds until the system velocity comes to zero at 2tt :

0)( 2.

tu 8

tn

nt2 1

2 1

1

( )

9

)1(2

1)( 12

nutu

10

Eqs. 9 and 10 indicate that the response of the non-linear SDOF system will only remain

bounded if n is less than 1. In that case in phase three ( t t 2 ), an elastic unloading in the spring

takes place. The equation of equilibrium in this phase is:

0..

oy PPKuuM 11

With initial conditions from Eqs. 8 and 10, the solution of Eq. 11 becomes:

)1(2

124)]()(cos[1()(

2

21n

nnttnutu 12

Eq. 12 indicates that for a non-linear and bounded response (0.5

19

)](cos[)()1(2

12)( 2.11. ttuuu

n

nutu stst

13

Eq. 13 shows that the adapted response of a non-linear SDOF system subjected to a step load

can be divided into three distinct components. The first one, an elastic time independent

displacement (ust.), is equal to the response of the system to a static load of Po and

monotonically increases with an increase in the applied external step load (Po). The second

component, the plastic time independent displacement [u1(2n-1)/2/(1-n)], is zero when n 0.5 or

Po Py/2 (a pure elastic response) and becomes infinitive when n=1 or Po=Py (a dynamic failure).

The magnitude of this plastic displacement depends on the coefficient n (the ratio between the

applied load and the plastic load) and monotonically increases with an increase in the applied

external step load (Po). The third component is the transient time dependent deformation

response )](cos[)( 2.1 ttuu st . This oscillation has its maximum amplitude of u1 when

n=0.5 (maximum elastic response). In post elastic response, the amplitude of the oscillations

decreases monotonically with increases in the applied impact load. With n=1. (a dynamic failure)

the amplitude of the elastic adaptation becomes equal to zero.

Displacement time histories of the non-linear SDOF model subjected to step impact loads are

shown in Fig. 11. The characteristics of the spring model (, Py, K and Po) have been adapted to

represent those from the previously studied individual tube (see Fig. 5). The displacements in

Fig. 11 are non-dimensional. The figure displays the purely elastic (Po=0.50Py) and the elasto-

plastic responses (with Po=0.80, 0.90, 0.935 and 0.95 Py). Fig. 11 shows similar behaviour as

presented in Figs. 5, 6 and 8. It can be seen that under these excitations the system remains

dynamically stable but the non-linear SDOF system eventually elastically shakes down. The

amplitude of the adopted elastic responses diminishes away as the system approaches its

dynamic failure limit (Po= Py). These responses can also be clearly recognized from Figs. 12 to

14.

Fig. 12 gives plots for the (non-dimensional) displacements versus the velocities in the nonlinear

SDOF system subjected to increasing step impact loads. Once again, when Po≤0.50Py the

system remains elastic, For Po>Py the response becomes unbounded and a dynamic failure

happens. With 0.50Py ≤Po≤Py, as it can be seen that large plastic displacements are developed

20

in the system, however, they end up in an adaptation phase. In this range, as the level of applied

load increases, the amplitude of the oscillations of these elastically shaken down responses

grow smaller.

0

0.5

1.0

1.5

0 50 100 150 200

P0 = 0.95 P

y

P0 = 0.935 P

y

P0 = 0.9 P

y

P0 = 0.8 P

y

P0 = 0.5 P

yp

TIME (ms)

DIS

PL

AC

EM

EN

T /

R

Fig. 11: Time history of the displacement in the non-linear SDOF model subjected to a step load

of P0.

-0.4

0

0.4

0.8

1.2

0 2 4 6 8 10

P0 = 0.6 P

yP

0 = 0.8 P

yP

0 = 0.9 P

yP

0 = 0.935 P

yP

0 = 0.95 P

y

Non-Dimensional Displacement (u/u1)

Non

-Dim

ensi

on

al

Vel

oci

ty(v

/u

1)

Fig. 12: Elastic shakedown/adaptation and degradation of the elastic response in a nonlinear

SDOF system subjected to increasing step loads of P0.

21

Fig. 13 shows plots of the (non-dimensional) velocities of the SDOF system subjected to

increasing step loads against its accelerations. Zones for the purely elastic behaviour, dynamic

failure and elastic shakedown are also shown (for the first quarter of the coordinate system).

When the external load grows closer to the failure point, adaptation and degradation of the

elastic responses can be recognized.

It should be mentioned that with a step load type excitation, the dynamic failure in the non-linear

SDOF system will be characterised by unbounded responses, so there will be no ratcheting type

failure. With other excitations, e.g. a harmonic type, a nonlinear SDOF system may experience

alternating plasticity or a ratcheting type of failure [23]. As it was already mentioned, no damping

was considered so the elastically shaken responses have closed loop forms (Figs. 12, and 13).

If damping is introduced in the system, these closed loops will turn into spirals in which the loop

size steadily decreases until it vanishes.

-0.5

0

0.5

1.0

-0.5 0 0.5 1.0

Elasticity

Elastic Shakedown

Unbounded ResponseP

0 = P

y

P0 = 0.5 P

y

P0 = 0.9 P

y

P0 = 0.8 P

y

P0 = 0.6 P

y

P0 = 0.95 P

y

Non-Dimensional Velocity (v/u1)

Non

-Dim

ensi

on

al

Acc

eler

ati

on

(a/

2u

1)

Fig. 13: Elastic shakedown/adaptation and degradation of the elastic response in a nonlinear

SDOF system subjected to increasing step loads of P0.

Fig. 14 gives the load displacement curves for the nonlinear spring in the SDOF model

subjected to increasing external step excitations of P0. The occurrence of adaptation and elastic

shakedown can be obviously recognized from the figure (see also Fig. 1b). After undergoing

nonlinear behaviour the system behaves in purely elastic cycles of loading and unloading. The

22

range of these cycles decreases as the excitation to the SDOF system approaches the dynamic

failure point.

0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12

P0 = 0.6 P

y

P0 = 0.8 P

y

P0 = 0.9 P

y

P0 = 0.95 P

y

P0 = 0.935 P

y

Non-Dimensional Displacement (u/u1)

Sp

rin

g F

orc

e /

Py

Fig. 14: Load displacement curves for the nonlinear spring in the SDOF model subjected to a

step load of P0.

With the SDOF models studied, the external load had no cyclic nature (with no repetition),

however, the impacted systems presented behaviours similar to the elastic shakedown of

structures which basically happens under cyclic loads. This is because in a dynamically excited

structure, even with time independent external loading, the structure becomes subject to cycles

of acceleration (inertia), velocity (damping), and displacement (stiffness) based on internal

loading and unloading. With this internal cyclic loading, the structure could become prone to

various types of shakedown.

The rationale provided in this section has been outlined for a time independent external load.

In the SDOF models studied in this section an elastic perfectly plastic behaviour has been

considered for the system. The study may be further enriched by integrating plastic hardening

(isotropic, kinematic or mixed hardening), as studied by Savi and Pacheco [24], or extending it to

MDOF systems. The external load has had a time independent nature. The study may be

extended to examining the adaptation, degradation of the adapted response, alternating

plasticity, ratcheting and dynamic failures in both the SDOF and MDOF systems subjected to

time varying excitations.

23

4. CLOSING REMARKS

This paper is devoted to some observations on the response of laterally impacted steel tubes,

which in some respects have been considered to be similar to shakedown. In these experiments

on steel tubes subjected to combinations of axial compression and lateral impacts, three

distinctive and interesting behaviours were noticed. With one response, after the development of

plastic deformation, the tubes ceased to exhibit further plastic deformation and reverted to a

purely elastic response. With the second behaviour, the amplitude of the elastic oscillations

became further restricted when the applied load approached the dynamic failure limits. Very

close to the dynamic limit load, the impacted tube only exhibited plastic deformation with no

perceptible elastic reactions and the amplitude of the elastic oscillations almost decreased to

zero.

Additional numerical and analytical investigations have been carried out on impacted tubes,

frames and non-linear SDOF systems to further examine the experimental observations. These

models have indicated similar results. They have substantiated that outside of the purely

elastic/unbounded response zones, the models studied exhibit behaviour similar to elastic

shakedown and adaptation. This means that although the impacted structure experiences

nonlinear deformations, the asymptotic responses remain elastic. The amplitude of these

adopted elastic oscillations becomes more restricted with increases in the level of impact load.

With responses close to the dynamic limit load, the oscillations almost die away.

Degradation of the elastic reactions in the response of the dynamically exited systems could

have important effects on the post damage behaviour or the failure of structures subjected to

dynamic loads. In the current study the issue of the adaptation and degradation of elastic

reactions have been studied for systems subjected to impact loads. Considering up the subject

for other structures and other types of dynamic excitations remains subject to further

investigation.

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24

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